JEE Main & Advanced Mathematics Triangles & Properties of Triangle Question Bank Self Evaluation Test - Properties of Triangles and Height & Dstances

In a triangle , if \[{{r}_{1}}=2{{r}_{2}}=3{{r}_{3}},\] then \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\] is equal to

A) \[\frac{75}{60}\]

B) \[\frac{155}{60}\]

C) \[\frac{176}{60}\]

D) \[\frac{191}{60}\]

Correct Answer: D

Solution :

[d] Given that, \[{{r}_{1}}=2{{r}_{2}}=3{{r}_{3}}\]

\[\therefore \,\,\,\,\frac{\Delta }{s-a}=\frac{2\Delta }{s-b}=\frac{3\Delta }{s-c}=\frac{\Delta }{k}\]

Then, \[s-a=k,s-b=2k,s-c=3k\]

\[\Rightarrow 3s-(a+b+c)=6k\Rightarrow s=6k\]

\[\therefore \frac{a}{5}=\frac{b}{4}=\frac{c}{3}=k\]

Now, \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=\frac{5}{4}+\frac{4}{3}+\frac{3}{5}\]

\[=\frac{75+80+36}{60}=\frac{191}{60}\]